On a New Construction of Generalized q-Bernstein Polynomials Based on Shape Parameter λ

This paper deals with several approximation properties for a new class of q-Bernstein polynomials based on new Bernstein basis functions with shape parameter λ on the symmetric interval [−1,1]. Firstly, we computed some moments and central moments. Then, we constructed a Korovkin-type convergence theorem, bounding the error in terms of the ordinary modulus of smoothness, providing estimates for Lipschitz-type functions. Finally, with the aid of Maple software, we present the comparison of the convergence of these newly constructed polynomials to the certain functions with some graphical illustrations and error estimation tables.

A Class of Sextic Trigonometric Bézier Curve with Two Shape Parameters

In this paper, we present a new class of sextic trigonometric Bernstein (ST-Bernstein, for short) basis functions with two shape parameters along with their geometric properties which are similar to the classical Bernstein basis functions. A sextic trigonometric Bézier (ST-Bézier, for short) curve with two shape parameters and their geometric characteristics is also constructed. The continuity constraints for the connection of two adjacent ST-Bézier curves segments are discussed. Shape control parameters can provide an opportunity to modify the shape of curve as designer desired. Some open and closed curves are also part of this study.

A Generalized Quasi Cubic Trigonometric Bernstein Basis Functions and Its B-Spline Form

In this paper, under the framework of Extended Chebyshev space, four new generalized quasi cubic trigonometric Bernstein basis functions with two shape functions α(t) and β(t) are constructed in a generalized quasi cubic trigonometric space span{1,sin2t,(1−sint)2α(t),(1−cost)2β(t)}, which includes lots of previous work as special cases. Sufficient conditions concerning the two shape functions to guarantee the new construction of Bernstein basis functions are given, and three specific examples of the shape functions and the related applications are shown. The corresponding generalized quasi cubic trigonometric Bézier curves and the corner cutting algorithm are also given. Based on the new constructed generalized quasi cubic trigonometric Bernstein basis functions, a kind of new generalized quasi cubic trigonometric B-spline basis functions with two local shape functions αi(t) and βi(t) is also constructed in detail. Some important properties of the new generalized quasi cubic trigonometric B-spline basis functions are proven, including partition of unity, nonnegativity, linear independence, total positivity and C2 continuity. The shape of the parametric curves generated by the new proposed B-spline basis functions can be adjusted flexibly.

Matrix representations for a certain class of combinatorial numbers associated with Bernstein basis functions and cyclic derangements and their probabilistic and asymptotic analyses

In this paper, we mainly concerned with an alternate form of the generating functions for a certain class of combinatorial numbers and polynomials. We give matrix representations for these numbers and polynomials with their applications. We also derive various identities such as Rodrigues-type formula, recurrence relation and derivative formula for the aforementioned combinatorial numbers. Besides, we present some plots of the generating functions for these numbers. Furthermore, we give relationships of these combinatorial numbers and polynomials with not only Bernstein basis functions, but the two-variable Hermite polynomials and the number of cyclic derangements. We also present some applications of these relationships. By applying Laplace transform and Mellin transform respectively to the aforementioned functions, we give not only an infinite series representation, but also an interpolation function of these combinatorial numbers. We also provide a contour integral representation of these combinatorial numbers. In addition, we construct exponential generating functions for a new family of numbers arising from the linear combination of the numbers of cyclic derangements in the wreath product of the finite cyclic group and the symmetric group of permutations of a set. Finally, we analyse the aforementioned functions in probabilistic and asymptotic manners, and we give some of their relationships with not only the Laplace distribution, but also the standard normal distribution. Then, we provide an asymptotic power series representation of the aforementioned exponential generating functions.

Geometric Modeling of Novel Generalized Hybrid Trigonometric Bézier-Like Curve with Shape Parameters and Its Applications

The main objective of this paper is to construct the various shapes and font designing of curves and to describe the curvature by using parametric and geometric continuity constraints of generalized hybrid trigonometric Bézier (GHT-Bézier) curves. The GHT-Bernstein basis functions and Bézier curve with shape parameters are presented. The parametric and geometric continuity constraints for GHT-Bézier curves are constructed. The curvature continuity provides a guarantee of smoothness geometrically between curve segments. Furthermore, we present the curvature junction of complex figures and also compare it with the curvature of the classical Bézier curve and some other applications by using the proposed GHT-Bézier curves. This approach is one of the pivotal parts of construction, which is basically due to the existence of continuity conditions and different shape parameters that permit the curve to change easily and be more flexible without altering its control points. Therefore, by adjusting the values of shape parameters, the curve still preserve its characteristics and geometrical configuration. These modeling examples illustrate that our method can be easily performed, and it can also provide us an alternative strong strategy for the modeling of complex figures.

A Novel Generalization of Trigonometric Bézier Curve and Surface with Shape Parameters and Its Applications

Adopting a recurrence technique, generalized trigonometric basis (or GT-basis, for short) functions along with two shape parameters are formulated in this paper. These basis functions carry a lot of geometric features of classical Bernstein basis functions and maintain the shape of the curve and surface as well. The generalized trigonometric Bézier (or GT-Bézier, for short) curves and surfaces are defined on these basis functions and also analyze their geometric properties which are analogous to classical Bézier curves and surfaces. This analysis shows that the existence of shape parameters brings a convenience to adjust the shape of the curve and surface by simply modifying their values. These GT-Bézier curves meet the conditions required for parametric continuity (C0, C1, C2, and C3) as well as for geometric continuity (G0, G1, and G2). Furthermore, some curve and surface design applications have been discussed. The demonstrating examples clarify that the new curves and surfaces provide a flexible approach and mathematical sketch of Bézier curves and surfaces which make them a treasured way for the project of curve and surface modeling.

Curve and surface construction based on the generalized toric-Bernstein basis functions

The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.